A Fractal Dimension for Measures via Persistent Homology

We use persistent homology in order to define a family of fractal dimensions, denoted dimPHi for each homological dimension i ≥ 0, assigned to a probability measure μ on a metric space. The case of zero-dimensional homology (i = 0) relates to work by Steele (Ann Probab 16(4): 1767–1787, 1988) studying the total length of a minimal spanning tree on a random sampling of points. Indeed, if μ is supported on a compact subset of Euclidean space ℝ^m for m ≥ 2, then Steele’s work implies that dimPH0(μ)=m if the absolutely continuous part of μ has positive mass, and otherwise dimPH0(μ)<m. Experiments suggest that similar results may be true for higher-dimensional homology 0 < i < m, though this is an open question. Our fractal dimension is defined by considering a limit, as the number of points n goes to infinity, of the total sum of the i-dimensional persistent homology interval lengths for n random points selected from μ in an i.i.d. fashion. To some measures μ, we are able to assign a finer invariant, a curve measuring the limiting distribution of persistent homology interval lengths as the number of points goes to infinity. We prove this limiting curve exists in the case of zero-dimensional homology when μ is the uniform distribution over the unit interval, and conjecture that it exists when μ is the rescaled probability measure for a compact set in Euclidean space with positive Lebesgue measure.